4.7. NATURAL CONVECTION HEAT TRANSFER

4.7A. Introduction

Natural convection heat transfer occurs when a solid surface is in contact with a gas or liquid which is at a different temperature from the surface. Density differences in the fluid arising from the heating process provide the buoyancy force required to move the fluid. Free or natural convection is observed as a result of the motion of the fluid. An example of heat transfer by natural convection is a hot radiator used for heating a room. Cold air encountering the radiator is heated and rises in natural convection because of buoyancy forces. The theoretical derivation of equations for natural convection heat-transfer coefficients requires the solution of motion and energy equations.

An important heat-transfer system occurring in process engineering is that in which heat is being transferred from a hot vertical plate to a gas or liquid adjacent to it by natural convection. The fluid is not moving by forced convection but only by natural or free convection. In Fig. 4.7-1 the vertical flat plate is heated and the free convection boundary layer is formed. The velocity profile differs from that in a forced convection system in that the velocity at the wall is zero and also is zero at the other edge of the boundary layer, since the free stream velocity is zero for natural convection. The boundary layer initially is laminar as shown, but at some distance from the leading edge it starts to become turbulent. The wall temperature is Tw K and the bulk temperature Tb.

Figure 4.7-1. Boundary-layer velocity profile for natural convection heat transfer from a heated, vertical plate.


The differential momentum-balance equation is written for the x and y directions for the control volume (dx dy · 1). The driving force is the buoyancy force in the gravitational field and is due to the density difference of the fluid. The momentum balance becomes

Equation 4.7-1


where ρb is the density at the bulk temperature Tb and ρ the density at T. The density difference can be expressed in terms of the volumetric coefficient of expansion β and substituted back into Eq. (4.7-1):

Equation 4.7-2


For gases, β = 1/T. The energy-balance equation can be expressed as follows:

Equation 4.7-3


The solutions of these equations have been obtained by using integral methods of analysis discussed in Section 3.10. Results have been obtained for a vertical plate, which is the simplest case and serves to introduce the dimensionless Grashof number discussed below. However, in other physical geometries the relations are too complex and empirical correlations have been obtained. These are discussed in the following sections.

4.7B. Natural Convection from Various Geometries

1. Natural convection from vertical planes and cylinders

For an isothermal vertical surface or plate with height L less than 1 m (P3), the average natural convection heat-transfer coefficient can be expressed by the following general equation:

Equation 4.7-4


where a and m are constants from Table 4.7-1, NGr the Grashof number, ρ density in kg/m3, μ viscosity in kg/m · s, ΔT the positive temperature difference between the wall and bulk fluid or vice versa in K, k the thermal conductivity in W/m · K, cp the heat capacity in J/kg · K, β the volumetric coefficient of expansion of the fluid in 1/K [for gases β is 1/(TfK)], and g is 9.80665 m/s2. All the physical properties are evaluated at the film temperature Tf = (Tw + Tb)/2. In general, for a vertical cylinder with length L m, the same equations can be used as for a vertical plate. In English units β is 1/(Tf °F + 460) in 1/°R and g is 32.174 × (3600)2 ft/h2.

Table 4.7-1. Constants for Use with Eq. (4.7-4) for Natural Convection
Physical GeometryNGrNPramRef.
Vertical planes and cylinders    
 [vertical height L < 1 m (3 ft)]    
  <1041.36(P3)
  104-1090.59(M1)
  >1090.13(M1)
Horizontal cylinders    
 [diameter D used for L and D < 0.20 m (0.66 ft)]    
  <10-50.490(P3)
  10-5-10-30.71(P3)
  10-3-11.09(P3)
  1-1041.09(P3)
  104-1090.53(M1)
  >1090.13(P3)
Horizontal plates    
 Upper surface of heated plates or ower surface of cooled plates105-2 × 107 2 × 107-3 × 10100.54 0.14 (M1) (M1)
 Lower surface of heated plates or upper surface of cooled plates105-10110.58(F1)

The Grashof number can be interpreted physically as a dimensionless number that represents the ratio of the buoyancy forces to the viscous forces in free convection and plays a role similar to that of the Reynolds number in forced convection.

EXAMPLE 4.7-1. Natural Convection from Vertical Wall of an Oven

A heated vertical wall 1.0 ft (0.305 m) high of an oven for baking food with the surface at 450°F (505.4 K) is in contact with air at 100°F (311 K). Calculate the heat-transfer coefficient and the heat transfer/ft (0.305 m) width of wall. Note that heat transfer for radiation will not be considered. Use English and SI units.

Solution: The film temperature is


The physical properties of air at 275°F are k = 0.0198 btu/h · ft · °F, 0.0343 W/m · K; ρ = 0.0541 lbm/ft3, 0.867 kg/m3; NPr = 0.690; μ = (0.0232 cp) × (2.4191) = 0.0562 lbm/ft · h = 2.32 × 10-5 Pa · s; β = 1/408.2 = 2.45 × 10–3 K-1, β = 1/(460 + 275) = 1.36 × 10–3 °R-1; ΔT = Tw - Tb = 450 - 100 = 350°F (194.4 K). The Grashof number is, in English units,


In SI units,


The Grashof numbers calculated using English and SI units must, of course, be the same as shown:


Hence, from Table 4.7-1, a = 0.59 and m = for use in Eq. (4.7-4). Solving for h in Eq. (4.7-4) and substituting known values,


For a 1-ft width of wall, A = 1 × 1 = 1.0 ft2 (0.305 × 0.305 m2). Then


A considerable amount of heat will also be lost by radiation. This will be considered in Section 4.10.


Simplified equations for the natural convection heat transfer from air to vertical planes and cylinders at 1 atm abs pressure are given in Table 4.7-2. In SI units the equation for the range of NGrNPr of 104 to 109 is the one usually encountered, and this holds for (L3 ΔT) values below about 4.7 m3 · K and film temperatures between 255 and 533 K. To correct the value of h to pressures other than 1 atm, the values of h in Table 4.7-2 can be multiplied by (p/101.32)1/2 for NGrNPr 104 to 109 and by (p/101.32)2/3 for NGrNPr > 109, where p = pressure in kN/m2. In English units the range of NGrNPr of 104 to 109 is encountered when (L3 ΔT) is less than about 300 ft3 · °F. The value of h can be corrected to pressures other than 1.0 atm abs by multiplying the h at 1 atm by p1/2 for NGrNPr of 104 to 109 and by p2/3 for NGrNPr above 109, where p = atm abs pressure. Simplified equations are also given for water and organic liquids.

Table 4.7-2. Simplified Equations for Natural Convection from Various Surfaces
   EquationRef.
Physical GeometryNGrNPrh = btu/h · ft2 · °F

L = ft, ΔT = °F

D = ft
h = W/m2 · K

L = m, ΔT = K

D = m
Air at 101.32 pa (1 atm) abs pressure
Vertical planes and cylinders104-109 >109h = 0.28(ΔT/L)1/4 h = 0.18(ΔT)1/3h = 1.37(ΔT/L)1/4 h = 1.24 ΔT1/3(P1) (P1)
Horizontal cylinders103-109 >109h = 0.27(ΔT/D)1/4 h = 0.18(ΔT)1/3h = 1.32(ΔT/D)1/4 h = 1.24 ΔT1/3(M1) (M1)
Horizontal plates    
 Heated plate facing upward or cooled plate facing downward105-2 × 107 2 × 107-3 × 1010h = 0.2T(ΔT/L)1/4 h = 0.22(ΔT)1/3h = 1.32(ΔT/L)1/4 h = 1.52 ΔT1/3(M1) (M1)
 Heated plate facing downward or cooled plate facing upward3 × 105-3 × 1010h = 0.12(ΔT/L)1/4h = 0.59(ΔT/L)1/4(M1)
Water at 70°F (294 K)
Vertical planes and cylinders104-109h = 26(ΔT/L)1/4h = 12T(ΔT/L)1/4(P1)
Organic liquids at 70°F (294 K)
Vertical planes and cylinders104-109h = 12(ΔT/L)1/4h = 59(ΔT/L)1/4(P1)

EXAMPLE 4.7-2. Natural Convection and Simplified Equation

Repeat Example 4.7-1 but use the simplified equation.

Solution: The film temperature of 408.2 K is in the range 255–533 K. Also,


This is slightly greater than the value of 4.7 given as the approximate maximum for use of the simplified equation. However, in Example 4.7-1 the value of NGrNPr is below 109, so the simplified equation from Table 4.7-2 will be used:


The heat-transfer rate q is


This value is reasonably close to the value of 127.1 W for Example 4.7-1.


2. Natural convection from horizontal cylinders

For a horizontal cylinder with an outside diameter of D m, Eq. (4.7-4) is used with the constants given in Table 4.7-1. The diameter D is used for L in the equation. Simplified equations are given in Table 4.7-2. The usual case for pipes is for the NGrNPr range 104 to 109 (M1).

3. Natural convection from horizontal plates

For horizontal flat plates Eq. (4.7-4) is also used with the constants given in Table 4.7-1 and simplified equations in Table 4.7-2. The dimension L to be used is the length of a side of a square plate, the linear mean of the two dimensions for a rectangle, and 0.9 times the diameter of a circular disk.

4. Natural convection in enclosed spaces

Free convection in enclosed spaces occurs in a number of processing applications. One example is in an enclosed double window in which two layers of glass are separated by a layer of air for energy conservation. The flow phenomena inside these enclosed spaces are complex, since a number of different types of flow patterns can occur. At low Grashof numbers the heat transfer is mainly by conduction across the fluid layer. As the Grashof number is increased, different flow regimes are encountered.

The system for two vertical plates of height L m containing the fluid with a gap of δ m is shown in Fig. 4.7-2, where the plate surfaces are at temperatures T1 and T2. The Grashof number is defined as

Equation 4.7-5


Figure 4.7-2. Natural convection in enclosed vertical space.


The Nusselt number is defined as

Equation 4.7-6


The heat flux is calculated from

Equation 4.7-7


The physical properties are all evaluated at the mean temperature between the two plates. For gases enclosed between vertical plates and L/δ > 3 (H1, J1, K1, P1),

Equation 4.7-8


Equation 4.7-9


Equation 4.7-10


For liquids in vertical plates,

Equation 4.7-11


Equation 4.7-12


For gases or liquids in a vertical annulus, the same equations hold as for vertical plates.

For gases in horizontal plates with the lower plate hotter than the upper,

Equation 4.7-13


Equation 4.7-14


For liquids in horizontal plates with the lower plate hotter than the upper (G5),

Equation 4.7-15


EXAMPLE 4.7-3. Natural Convection in Enclosed Vertical Space

Air at 1 atm abs pressure is enclosed between two vertical plates where L = 0.6 m and δ = 30 mm. The plates are 0.4 m wide. The plate temperatures are T1 = 394.3 K and T2 = 366.5 K. Calculate the heat-transfer rate across the air gap.

Solution: The mean temperature between the plates is used to evaluate the physical properties: Tf = (T1 + T2)/2 = (394.3 + 366.5)/2 = 380.4 K. Also, δ = 30/1000 = 0.030 m. From Appendix A.3, ρ = 0.9295 kg/m3, μ = 2.21 × 10-5 Pa · s, k = 0.03219 W/m · K, NPr = 0.693, β = 1/Tf = 1/380.4 = 2.629 × 10–3 K-1.




5. Natural convection from other shapes

For spheres, blocks, and other types of enclosed air spaces, references elsewhere (H1, K1, M1, P1, P3) should be consulted. In some cases when a fluid is forced over a heated surface at low velocity in the laminar region, combined forced convection plus natural convection heat transfer occurs. For further discussion of this, see (H1, K1, M1).

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